Critical power (CP) theoretically represents the highest power output that can be maintained indefinitely or at least for a very long time without exhaustion and demarcates the heavy and severe domains of exercise intensity (^{10,20}). Because of the pronounced slow component of V˙O_{2} in the severe-intensity domain (^{10}), it has been shown that V˙O_{2} can increase to its maximum for power outputs greater than CP (^{6,11–13,20,29,30}). However, this has been shown only when CP has been derived from linearized formulations of the hyperbolic relationship between power and time to exhaustion (TTE). Although some have reported similar CP estimates with linear and nonlinear models (^{7,21,25}), we (^{9}) and others (^{3,4,16}) have previously reported that parameter estimates of the hyperbolic power–endurance time relationship are dependent upon the model used and that for cycle ergometry, nonlinear two- and three-parameter models produce estimates of CP that are ∼15–40 W lower than those produced with linear models (^{3,9}). Thus, it is unclear whether exercise in the severe-intensity domain invariably results in V˙O_{2} rising to its maximum.

The purpose of this study was to determine whether V˙O_{2max} is attained during exercise to exhaustion at power outputs greater than CP. We hypothesized that V˙O_{2max} would not be attained during exercise to exhaustion for all power outputs greater than CP, particularly those just above CP (i.e., at the lower range of the severe-intensity domain).

## METHODS

#### Subjects.

Nine physically active males and six females between 21 and 39 yr volunteered to be subjects. Each subject was informed of all the risks associated with participation in the study and gave written consent to participate. The study was approved by the human investigation committees of both the University of Virginia and Longwood College, where two of the investigators (Gaesser and Womack) had appointments before their current positions. Two of the female subjects’ data were eliminated from this analysis because of inconsistent V˙O_{2} values on constant-power tests. Therefore, our analysis included data from nine males and four females. The means ± SD for age, height, weight, and body mass index are shown in Table 1.

#### Determination of V˙O_{2max}.

Subjects first performed an incremental exercise test (IET) on a Monark friction-braked cycle ergometer (Monark AB, Vansbro, Sweden) to determine V˙O_{2max}. Throughout the test, subjects were instructed to maintain pedaling cadence at 60 rpm with the assistance of a calibrated metronome. Cadence and load setting on the ergometer were continuously monitored by a laboratory technician. The test started with 2 min of unloaded cycling. Thereafter, power output was increased by 30 W each minute (males) or 15 W each minute (females) until volitional exhaustion. Subjects were given strong verbal encouragement to maintain the desired cadence. Once cadence fell below 50 rpm and despite strong verbal encouragement to increase cadence to the desired 60 rpm, there was typically a precipitous drop in pedaling frequency with complete inability to move the pedals within 10–15 s. The highest power achieved during this incremental test was used as a reference point for the selection of power outputs to be used during subsequent constant-power exercise bouts to exhaustion (^{5,9}). Pulmonary ventilation and gas exchange were continuously measured either with a SensorMedics 2900 metabolic measurement cart (SensorMedics Corp, Yorba Linda, CA) (five subjects at Longwood College) or as previously described using a calibrated dry gas meter (Rayfield RAM-9200, Rayfield Equipment Ltd., Waitsfield, VT) fitted with a potentiometer, 7-L mixing chamber, and Applied Electrochemistry S-3A oxygen analyzer (AEI Technologies, Applied Electrochemistry, Pittsburgh, PA) and Beckman LB-2 carbon dioxide analyzer (Beckman, Schiller Park, IL) (eight subjects at the University of Virginia) (^{35,38}). V˙O_{2max} was taken as the highest 15-s value reached during the IET.

#### Determination of the power–endurance relationship.

Four constant-power outputs were selected as described previously (^{5,9}) and were designed to produce exercise TTE in the range of 1–20 min. Subjects completed these four tests in random order on separate days. Each test was separated by at least 48 h. If the lowest power output resulted in a TTE <10 min, a fifth exercise bout was performed at a lower power output. After a warm-up on the same cycle ergometer used for the IET, subjects pedaled to exhaustion at a cadence of their choice between 60 and 90 rpm (all subjects chose 60 or 70 rpm, except two, who chose 80 and 90 rpm). Pedaling frequency was achieved by having subjects pedal to a calibrated metronome while having cadence monitored by a technician. The load setting on the ergometer was continuously monitored and adjusted, as needed, to keep power output constant (^{5}). Exhaustion was defined as the point at which the subject could no longer maintain his or her chosen pedaling cadence. When cadence fell below the desired revolutions per minute, the subject was given verbal encouragement to pedal faster. If the subject could not return to the required revolutions per minute, the test was terminated, and TTE was recorded to the nearest second. At no time was the subject provided information about the elapsed time or power output of the test. Pulmonary ventilation and gas exchange were continuously measured as described for the IET.

#### Mathematical modeling.

As described previously (^{9}), we used the following models to produce parameter estimates of the power–duration relationship:

where *P* = power, *t* = TTE, *W* = work (*Pt*), *W*′ represents the curvature constant of the power–duration hyperbola, and CP represents the highest sustainable aerobic power output. In the three-parameter model, a third term, *k* [*W*′ / (*P*_{max} − CP); see Jones et al. (^{20}) and Morton (^{24}) for further discussion], is added to allow for estimation of a finite intercept of *t* on the *P* axis when the dependent variable *t* is plotted as a function of the independent variable *P* (^{9,20,24}). *P*_{max} essentially represents maximal instantaneous power (^{9,20,24}).

#### Statistical analysis.

Statistical analysis was performed on PASW™ (SPSS, Chicago, IL) using linear mixed models with gender within subjects as the random factor and V˙O_{2} as the fixed factor to compare mean V˙O_{2} values for each constant-power test with V˙O_{2max} as well as CP values estimated by different modeling techniques. Bonferroni *post hoc* tests were used to find which V˙O_{2} value differed from V˙O_{2max} and which CP estimate differed from the others. The parameter estimates and 95% confidence intervals (CI) were derived using linear and nonlinear regression in PASW™. The CP estimates were each checked for interrelationships using Pearson correlations within subjects. An *α* level of *P* < 0.05 was used to determine statistical significance.

## RESULTS

Each subject performed a minimum of four constant-power exercise tests (*P*_{1} = highest, *P*_{4} = lowest). Three of the 13 subjects completed a fifth constant-power test (*P*_{5}) at a lower power output than *P*_{4} to ensure that they had one exercise bout with a TTE ∼15–20 min. All but four of the 55 total constant-power exercise tests resulted in a TTE between 1 and 20 min, with three subjects completing approximately 20–23 min and one completing 30 min at the lowest power output (Table 2). Therefore, the *P*_{4/5} data set represents the lowest power output for each of the 13 subjects.

The IET V˙O_{2max} averaged 3.55 ± 0.92 L·min^{−1} (RER = 1.21 ± 0.05, HR = 186 ± 10 bpm, 96.1% ± 6.3% of age-predicted maximum; Table 1). All subjects reached an RER of at least 1.15 during the IET, and 11 of the 13 subjects reached an HR_{max} of at least 90% of age-predicted maximum. The remaining two subjects reached 87% and 88% of the age-predicted HR_{max}, but their RER_{max} values were 1.29 and 1.22, indicative of a maximal effort [see Howley et al. (^{17}), but also see Poole et al. (^{31})].

The mixed-model analysis showed there was a significant difference in V˙O_{2peak} values attained during the constant-power exercise bouts and IET V˙O_{2max} (*F* = 5.15, *P* = 0.001; Table 2 and Fig. 1). These V˙O_{2peak} values ranged from 3.11 ± 0.79 L·min^{−1} (*P*_{4/5}) to 3.54 ± 0.91 L·min^{−1} (*P*_{2}) (Table 2). Bonferroni *post hoc* tests showed that V˙O_{2peak} values attained during the three highest constant-power bouts (*P*_{1} = 3.32 ± 0.88 L·min^{−1}, *P*_{2} = 3.54 ± 0.91 L·min^{−1}, *P*_{3} = 3.44 ± 0.91 L·min^{−1}) were not significantly different (*P* = 0.14, 0.47, and 0.24 for *P*_{1}, *P*_{2}, and *P*_{3}, respectively) from IET V˙O_{2max} (3.55 ± 0.92 L·min^{−1}). These end-exercise V˙O_{2} values represented 96.5% ± 10.3%, 100.7% ± 11.2%, and 97.0% ± 9.8% of IET V˙O_{2max}, for *P*_{1}, *P*_{2}, and *P*_{3}, respectively. However, V˙O_{2peak} during the lowest constant-power exercise bout for all 13 subjects (either *P*_{4} or *P*_{5}, represented by *P*_{4/5}; 3.11 ± 0.79 L·min^{−1}) was significantly lower than IET V˙O_{2max} (*P* = 0.004; Table 2). The end-exercise V˙O_{2} for *P*_{4/5} represented only 88.2% ± 9.4% V˙O_{2max} (Fig. 1).

The V˙O_{2} profile for the *P*_{4/5} exercise bout revealed a slow component of V˙O_{2} with a delayed steady state after ∼60% of the total exercise time, which averaged 18.48 ± 4.43 min (range = 14.63–30.18 min) (Fig. 2). The regression of V˙O_{2} on time during the last 40% of the total exercise time (approximately the last 6–12 min of each test) for each subject revealed, in all instances, a slope not significantly different from zero. Thus, a plateau in V˙O_{2} was demonstrated for each of the 13 subjects during the *P*_{4/5} constant-power exercise bout.

The mixed-model analysis showed there was a significant difference between CP estimates (*F* = 56.85, *P* < 0.001; Table 3). Mean CP estimates ranged from 159 ± 40 W for the three-parameter nonlinear model to 186 ± 43 for the linear (*P*) model. For all 13 subjects, the three-parameter nonlinear model produced the lowest estimate of CP, and the linear (*P*) model produced the highest estimate of CP. Despite all equations producing excellent goodness of fit to the power–endurance time data (*R*^{2} = 0.983–0.997; Table 3), Bonferroni *post hoc* tests revealed that estimates of CP were all significantly different from one another (all *P* values < 0.001; Table 3).

For each subject, *P*_{4/5} fell within the 95% CI for CP estimated by both linear models (mean data are presented in Table 3). By contrast, for the two-parameter nonlinear model, the *P*_{4/5} was above the 95% CI for all but three of the subjects. For the three subjects whose *P*_{4/5} fell within the 95% CI, the *P*_{4/5} for two of these subjects was within 2 W of the upper bound of the 95% CI, and one subject’s *P*_{4/5} was within 11 W of the upper bound of the 95% CI. Despite excellent goodness of fit similar to the other models, the three-parameter nonlinear model produced very large 95% CIs, roughly two to four times that of the other models (Table 3). Nevertheless, six of the subjects’ *P*_{4/5} fell above the upper bound of the 95% CI for the CP estimate.

## DISCUSSION

Our results indicate that exercise above CP does not inevitably cause oxygen uptake to rise to V˙O_{2max}, as postulated previously (^{6,10–13,20,29,30}). The lowest constant-power exercise bout (*P*_{4/5}) used to determine CP elicited a mean end-exercise V˙O_{2} of only 88.2% ± 9.4% IET V˙O_{2max}. When the highest V˙O_{2} during any of the constant-power exercise bouts is used for comparison, the *P*_{4/5} end-exercise V˙O_{2} elicited only 83.2% ± 7.0% of “maximum” V˙O_{2}. Furthermore, a clear plateau in V˙O_{2} was observed for the *P*_{4/5} exercise bout and was verified by regression analysis for each subject, supporting our hypothesis that not all exercise bouts in the severe domain elicit V˙O_{2max}.

Our results must be considered in the context of the CP estimates, including the 95% CI, from the different models. Only for the two-parameter nonlinear model was the power output during the *P*_{4/5} exercise bout above the upper bound of the 95% CI for the CP estimate for nearly all subjects. The fact that *P*_{4/5} could only be sustained for an average of 18.5 min is consistent with the notion that exercise time in the severe domain is finite and can be predicted on the basis of the power–duration hyperbolic relationship (^{20}). The results for the three-parameter model were mixed, in large part because of extremely large 95% CIs for CP estimation for several of the subjects (Table 3). The large 95% CIs are most likely due to the lower number of degrees of freedom in parameter estimation for the three-parameter model. Additional constant-power exercise bouts for parameter estimation could be expected to produce a narrower range of 95% CIs for the three-parameter model. For example, the mean ± SD range between the lower and upper bounds of the 95% CI was only 39 ± 11 W for the three subjects who completed a fifth constant-power exercise bout but was 160 ± 155 W for the 10 subjects who performed only four exercise bouts. Nevertheless, six subjects had power outputs for the *P*_{4/5} exercise bout that fell above the upper bound of the 95% CI for the three-parameter model. Three of these subjects reached 98% to 108% IET V˙O_{2max}, which is consistent with the notion that exercise greater than CP drives V˙O_{2} to maximum (^{6,10–13,20,29,30}). In contrast and in support of our hypothesis, V˙O_{2} for the other three subjects reached a plateau at only 73% to 84% of IET V˙O_{2max}. Additional research is needed to clarify the mixed results of the three-parameter nonlinear model and will require more constant-power exercise bouts to ensure confidence in parameter estimation (^{3,9}).

Our results are similar to those of Carter et al. (^{6}), who reported that constant-velocity treadmill runs at speeds well above critical velocity (CV, linear (*P*) model) led to the attainment of V˙O_{2max}, but running at a constant velocity just above CV did not result in V˙O_{2} reaching the maximum attained in the IET. Similar results for treadmill running exercise were also reported by Billat et al. (^{1}), who showed that well-trained runners could reach only 92.6% of IET V˙O_{2max} while exercising at a treadmill velocity 5% above CV. However, neither of these studies established 95% CIs for CP estimation.

Because *P*_{4/5} for all subjects fell within the 95% CI for CP estimation using the two linear models, we hesitate to draw any definitive conclusions about these models regarding our hypothesis. CP estimates were significantly higher for the two linear models (Table 3) and thus much closer to the *P*_{4/5} power output. The finding that the linear and nonlinear models of the power–duration hyperbola produced different estimates of CP is not surprising (^{3,4,9,16}), although some investigators have reported similar CP estimates with the two-parameter nonlinear and both linear models (^{7,21,25}). We have argued that nonlinear data should be treated with nonlinear statistical analyses and that the three-parameter nonlinear model is preferred because, in addition to appropriately designating *t* as the dependent variable, it also assumes that *P* is not infinite as *t* approaches zero (^{9}). Just as importantly, the three-parameter model produces a CP estimate that more closely reflects a power output (or velocity) that can be maintained for a very long time without exhaustion (^{4,9}).

Previous investigations have shown that the linear models overestimate CP (^{2,4,9,14,15,18,19,22,27,28,32}). Exercise at CP derived from either of the two linear models rarely can be maintained for longer than ∼30 min before exhaustion occurs (^{2,4,15,18,19,22,27,28}), with some studies reporting that exercise at CP could only be maintained for ∼15–25 min (^{4,15,27,28}). In the present study, exercise at 188 W (the mean of *P*_{4/5}) could be sustained for an average of only 18.5 min, yet this power output is essentially the same as the CP estimate (186 W) produced by the linear (*P*) model. By contrast, exercise at CP derived from the three-parameter nonlinear model can be maintained for at least 60 min by most subjects (^{3,4}). In addition, Gaesser et al. (^{9}) reported that CP estimated from the three-parameter model came closest to and was not statistically significantly different from the ventilation threshold for long-term exercise (^{33}). The ventilatory threshold for long-term exercise (^{33}) is similar to the maximum steady state (^{32}), which for cycle ergometry occurs at a power output below CP (^{32}). Therefore, the three-parameter nonlinear model should be used to estimate the power (or velocity) that more accurately reflects what it is theoretically supposed to represent (^{4,9,16,20}).

Our data show that exercise intensities in the severe domain, especially those that are well above CP, can drive V˙O_{2} to its maximum as previously reported (^{6,8,10–13,20,29–31,34,36,37,39}). The second and third highest power outputs (*P*_{2} = 247 ± 59 W, *P*_{3} = 218 ± 47 W) are significantly below the peak power associated with the IET V˙O_{2max} (300 ± 75 W) yet produced end-exercise V˙O_{2} values that were not significantly different from IET V˙O_{2max} (*P*_{2} = 100.7% ± 11.2% V˙O_{2max}, *P*_{3} = 97.0% ± 9.8% V˙O_{2max}). Close inspection of Figure 1, however, shows that there is considerable individual variability. Our data (Fig. 1) showing that “submaximal” exercise in the severe domain may elicit V˙O_{2peak} values greater than those achieved during a traditional IET suggest that an IET, even if accepted criteria for a maximum test are met (^{17}), may not elicit the true maximum for a particular subject (^{23}).

The observation that V˙O_{2peak} for *P*_{1} was not higher than that for either *P*_{2} or *P*_{3} is most likely due to the fact that the *P*_{1} power output (58 W higher than the mean peak power reached in the IET) was so high that it did not allow sufficient exercise time for V˙O_{2max} to be reached (^{13,26}). The average TTE for our subjects at *P*_{1} was only 1.25 min, with several subjects having a TTE of 54–65 s. During constant-load exercise to exhaustion, it may take >2 min for many individuals to reach V˙O_{2max} (^{13}).

Many researchers have recommended a verification exercise bout to establish whether a true V˙O_{2max} was reached during an IET (^{8,23,31,36}). Although we did not *a priori* include a verification test, the constant-power tests essentially served as *de facto* verification tests similar to previous investigations (^{8,31}). Poole et al. (^{31}) demonstrated the limitations of relying on secondary criteria, such as RER and percent of HR_{max} reached during the IET. Our data are in agreement with these reports by showing that a V˙O_{2} significantly higher than IET V˙O_{2max} may be attained during constant-power exercise bouts in the severe domain.

Primarily on the basis of the data from the two-parameter nonlinear model, we conclude that at power outputs above CP, V˙O_{2} does not necessarily increase to maximum during constant-power exercise to exhaustion. Furthermore, the highest V˙O_{2} measured during a traditional V˙O_{2} “max” test (IET) may not reflect the highest attainable V˙O_{2} even when V˙O_{2max} criteria are met. Further research should focus on exercise at CP and above CP (verified by confirming that the greater-than-CP power is above the upper bound of the 95% CI of the CP estimate) to clearly demarcate heavy and severe exercise domains. Also, continued research in the area of V˙O_{2max} should focus on methods similar to the verification phase to confirm “true” V˙O_{2max} (^{23}).

No funding was received for this research.

There are no conflicts of interest declared.

The results of the present study do not constitute endorsement by the American College of Sports Medicine.

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